An exterminator estimated there were 12,000 termites in a house. Each time the house was sprayed, the number of termites was reduced to one-fourth the previous number. How many termites were there after the house was sprayed [tex]$x$[/tex] times? Write a function to represent this scenario.
A. [tex]$f(x)=\frac{1}{4}(12,000)^x$[/tex]
B. [tex]$f(x)=12,000(\frac{1}{4})^x$[/tex]
C. [tex]$f(x)=12,000+(\frac{1}{4})^x$[/tex]
D. [tex]$f(x)=12,000-(\frac{1}{4})^x$[/tex]
Answer (2)
The problem describes an exponential decay scenario.
The general form of an exponential decay function is f ( x ) = a × b x , where a is the initial value and b is the decay factor.
In this case, the initial value is 12,000 and the decay factor is 1/4.
Therefore, the function is f ( x ) = 12 , 000 ( 4 1 ) x , so the answer is B .
Explanation
Problem Analysis Let's analyze the problem. We are given an initial number of termites (12,000) and a decay factor (1/4) for each time the house is sprayed. We need to find a function that represents the number of termites after x sprays.
Exponential Decay Function The general form of an exponential decay function is f ( x ) = a × b x , where a is the initial value and b is the decay factor. In this case, the initial value is 12,000 and the decay factor is 1/4.
Finding the Function Therefore, the function is f ( x ) = 12 , 000 × ( 1/4 ) x .
Comparing with Options Comparing this with the given options, we see that option B matches our function.
Final Answer The function that represents the number of termites after the house is sprayed x times is f ( x ) = 12 , 000 × ( 4 1 ) x . So the answer is B.
Examples
Exponential decay is a concept used in various real-life scenarios. For example, it can model the depreciation of a car's value over time. If a car initially costs $20,000 and loses 15% of its value each year, the car's value after x years can be modeled using an exponential decay function. Similarly, it can be used to model the decay of radioactive substances or the decrease in the concentration of a drug in the bloodstream over time. Understanding exponential decay helps in making informed decisions in finance, medicine, and environmental science.
The function that represents the number of termites after the house is sprayed x times is given by f ( x ) = 12 , 000 ( 4 1 ) x . This indicates an exponential decay function where each spray reduces the termite population to one-fourth of its previous count. The correct answer is option (B).
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